Search

Using quasiconformal surgery, we prove that any primitive, postcritically-finite hyperbolic polynomial can be tuned with an arbitrary generalized polynomial with non-escaping critical points, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves affirmatively a conjecture by Inou and Kiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable., To appear in Israel Journal of Mathematics

This work performs a direct comparison between generalized polynomial chaos (GPC) expansion techniques applied to structural acoustic problems. Broadly, the GPC techniques are grouped in two categories: , where the stochastic sampling is predetermined according to a quadrature rule; and , where an arbitrary selection of points is used as long as they are a representative sample of the random input. As a baseline comparison, Monte Carlo type simulations are also performed although they take many more sampling points. The test problems considered include both canonical and more applied cases that exemplify the features and types of calculations commonly arising in vibrations and acoustics. A range of different numbers of random input variables are considered. The primary point of comparison between the methods is the number of sampling points they require to generate an accurate GPC expansion. This is due to the general consideration that the most expensive part of a GPC analysis is evaluating the deterministic problem of interest; thus the method with the fewest sampling points will often be the fastest. Accuracy of each GPC expansion is judged using several metrics including basic statistical moments as well as features of the actual reconstructed probability density function.

This article presents and validates a new branch-and-bound algorithm for effectively solving the generalized polynomial problem (GPP). In this algorithm, a new affine relaxed technique is derived f...

Let K be a Makar-Limanov algebraically closed skew field. In the first part of this paper, we prove that the image of a generalized multilinear polynomial, with coefficients in K, evaluated over M m ( K ) , is M m ( K ) . In the second part, we show that any matrix in M m ( K ) may be written as the sum of three or fewer elements from the image of a generalized polynomial, with coefficients in K, evaluated over M m ( K ) .

We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees., 11 pages, 2 figures

In 2015, Guth proved that if $\EuScript{S}$ is a collection of $n$ $g$-dimensional semialgebraic sets in ${\mathbb{R}}^d$ and if $D\geq 1$ is an integer, then there is a $d$-variate polynomial $P$ ...

In this paper, several error bounds for the solution sets of the generalized polynomial complementarity problems (GPCPs) with explicit exponents are given. As the solution set of a GPCP is the solu...

This paper proposes generalized polynomial guidance for controlling the terminal velocity vector (speed and flight path angle), which could be critical for the effectiveness of the warhead of tactical ballistic missiles. A polynomial reference trajectory satisfying the initial and terminal altitudes and flight path angles is introduced with a guidance parameter that can be chosen to change the terminal speed. A single differential equation of the speed along the reference trajectory is then derived and an iterative search method for determining the guidance parameter to satisfy the prescribed terminal speed is proposed. Numerical simulation study with various impact angle and terminal speed constraints is conducted to demonstrate the performance of the proposed guidance method for terminal velocity control. Robustness of the proposed method to drag variations is also investigated to check the feasibility of generalized polynomial guidance for a practical purpose.

In this article, we investigate a stochastic Galerkin method for the Maxwell equations with random inputs. The generalized Polynomial Chaos (gPC) expansion technique is used to obtain a deterministic system of the gPC expansion coefficients. The regularity of the solution with respect to the random is analyzed. On the basis of the regularity results, the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved. Numerical examples are presented to support the theoretical analysis.

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula, we prove that in the unweighted case, these generalized Leja points recover the transfinite diameter., Comment: 15 pages, 2 figures. Substantial changes in Section 3 were required to correct an erroneous argument in Section 4 of the original version

In this paper, we investigate the generalized polynomial complementarity problem, which is a subclass of generalized complementarity problems with the involved map pairs being two polynomials. Based on the analysis on two structured tensor pairs located in the heading items of polynomials involved, and by using the degree theory, we achieve several results on the nonemptiness and compactness of solution sets. When generalized polynomial complementarity problems reduce to polynomial complementarity problems (or tensor complementarity problems), our results reduce to the existing ones. In particular, one of our results broadens the one proposed in a very recent paper to guarantee the nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. Furthermore, we establish several existence and uniqueness results, which enrich the theory of generalized complementarity problems with the observation that some known conditions to guarantee the existence and uniqueness of solutions may not hold for a lot of generalized polynomial complementarity problems.

The authors propose an algorithm to construct Chebyshev approximation for functions of several variables by a generalized polynomial as a limiting approximation in the norm of space Lp as p → ∞. It is based on serial construction of power-average approximations using the least squares method with variable weight function. The convergence of the method provides an original way to consistently refine the values of the weight function, which takes into account the results of approximation at all previous iterations. The authors describe the methods of calculating the Chebyshev approximation with absolute and relative errors. The results of test examples confirm the efficiency of using the method to obtain Chebyshev approximation of tabular continuous functions of one, two, and three variables.

Combined plant and control design (control co-design (CCD)) methods are often used during product development to address the synergistic coupling between the plant and control parts of a dynamic system. Recently, a few studies have started applying CCD to stochastic dynamic systems. In their most rigorous approach, reliability-based design optimization (RBDO) principles have been used to ensure solution feasibility under uncertainty. However, since existing reliability-based CCD (RBCCD) algorithms use all-at-once formulations, only most-probable-point (MPP) methods can be used as reliability analysis techniques. Though effective for linear/quadratic RBCCD problems, the use of such methods for highly nonlinear RBCCD problems introduces solution error that could lead to system failure. A multidisciplinary feasible (MDF) formulation for RBCCD problems would eliminate this issue by removing the dynamic equality constraints and instead enforcing them through forward simulation. Since the RBCCD problem structure would be similar to traditional RBDO problems, any of the well-established reliability analysis methods could be used. Therefore, in this work, a novel reliability-based MDF formulation of multidisciplinary dynamic system design optimization has been proposed for RBCCD. To quantify the uncertainty propagated by the random decision variables, Monte Carlo simulation has been applied to the generalized polynomial chaos expansion of the probabilistic constraints. The proposed formulation is applied to two engineering test problems, with the results indicating the effectiveness of both the overall formulation as well as the reliability analysis technique for RBCCD.

Objectives This paper is aimed at modelling the effect of COVID-19 mortality per population (CMP), a proxy for COVID-19 on the Gross Domestics Product (GDP) per capita per COVID-19 cases (RGDPC), a proxy for the economic wellbeing of a nation. Methods Nine models divided into three groups (Gaussian polynomial, other non-linear, and Gamma generalized polynomial models) were fitted for RGDPC data on CMP, collected from 1st June to 31st December 2020. Results The result showed that the gamma cubic model was selected as the best model out of the 9 competing models to predict the economic wellbeing of Nigeria. Predictions were made for the whole day in the year 2021. Conclusion It is therefore concluded that there is a non-linear relationship between COVID-19 mortality and the economic wellbeing of Nigerians. Thus, COVID-19 mortality has an adverse effect on the wellbeing of Nigerians. The economic wellbeing of Nigerians can be improved if COVID-19 mortality is stopped.

In this paper, we consider the generalized polynomial variational inequality, which is a subclass of generalized variational inequalities; and it covers several classes of variational inequalities with polynomial functions studied recently in the literature. A well-known existence and uniqueness theorem for the generalized variational inequality was established by Pang and Yao (SIAM J Control Optim 33:168–184, 1995). It is not difficult to show that the conditions of this theorem do not hold for generalized variational inequalities with general polynomial functions. In this paper, in terms of properties of the involved polynomial and by making use of the theory related to exceptional family of elements, we establish an existence and uniqueness theorem for the generalized polynomial variational inequality. A specific example is given to confirm our theoretical findings.

Many industrial and environmental processes are characterized as complex spatio-temporal systems. Such systems known as distributed parameter systems (DPSs) are usually highly complex and it is difficult to establish the relation between model inputs, model outputs and model parameters. Moreover, the solutions of physics-based models commonly differ somehow from the measurements. In this work, appropriate Uncertainty Quantification (UQ) approaches are selected and combined systematically to analyze and identify systems. However, there are two main challenges when applying the UQ approaches to nonlinear distributed parameter systems. These are: (1) how uncertainties are modeled and (2) the computational effort, as the conventional methods require numerous evaluations of the model to compute the probability density function of the response. This paper presents a framework to solve these two issues. Within the Bayesian framework, incomplete knowledge about the system is considered as uncertainty of the system. The uncertainties are represented by random variables, whose probability density function can be achieved by converting the knowledge of the parameters using the Principle of Maximum Entropy. The generalized Polynomial Chaos (gPC) expansion is employed to reduce the computational effort. The framework using gPC based on Bayesian UQ proposed in this work is capable of analyzing systems systematically and reducing the disagreement between model predictions and measurements of the real processes to fulfill user defined performance criteria. The efficiency of the framework is assessed by applying it to a benchmark model (neutron diffusion equation) and to a model of a complex rheological forming process. These applications illustrate that the framework is capable of systematically analyzing the system and optimally calibrating the model parameters.

In this paper we provide an upper bound on the maximum number of limit cycles for a class of generalized polynomial Lienard differential systems x = y , y = − f n ( x ) y − g m ( x ) with f n and g m real polynomials of degree n and m respectively, using the fourth order averaging method.

In this paper we develop a novel optimal control framework for uncertain mechanical systems. Our work extends differential dynamic programming and handles uncertainty through generalized polynomial...

A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomial chaos (GPC) method that approximates the stochastic response by orthogonal polynomials. The accuracy and effectiveness of the method are illustrated by different numerical examples including both linear and nonlinear problems. The results indicate that the proposed method modifies the stochastic bases adaptively, and has a better approximation for the probability density function in contrast to the GPC method.

Agraïments: This work is partially supported by grant CONACYT-58968. Applying the averaging theory of first, second and third order to one class generalized polynomial Li'enard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit.

The traditional fatigue life prediction methods based on the S-N curve all believe that the parameters in the model are deterministic constants and can be categorized to the deterministic life prediction. However, in practice, it is difficult to carry out a large number of experiments due to the limitation of time or the possible shortage of funds. In addition, the specimens used in the experiments are not exactly the same, and the test operations and data reading depend on the accuracy of the test equipment as well as the subjective judgment of the testers, which result to the uncertainty of the S-N curve. Therefore, the uncertainty should be considered in order to improve the accuracy of the fatigue life prediction. In this paper, the uncertain factors affecting the fatigue life of welded joints are summarized, and the generalized polynomial chaos (gPC) is introduced into fatigue life prediction. A novel probabilistic fatigue life prediction method combined with the nonlinear cumulative damage model considering the uncertainty of the S-N curve is constructed. An illustrative example is presented to demonstrate the advantages of the proposed approach.

In this paper, we consider the effect of stochastic uncertainties on non-linear systems with chaotic behavior. More specifically, we quantify the effect of parametric uncertainties to time-averaged quantities and their sensitivities. Sampling methods for Uncertainty Quantification (UQ), such as the Monte&ndash, Carlo (MC), are very costly, while traditional methods for sensitivity analysis, such as the adjoint, fail in chaotic systems. In this work, we employ the non-intrusive generalized Polynomial Chaos (gPC) for UQ, coupled with the Multiple-Shooting Shadowing (MSS) algorithm for sensitivity analysis of chaotic systems. It is shown that the gPC, coupled with MSS, is an appropriate method for conducting UQ in chaotic systems and produces results that match well with those from MC and Finite-Differences (FD).

The reliability analysis of a structural system is typically evaluated based on a multivariate model that describes the underlying mechanistic relationship between the system’s input and output random variables. This is the need to develop an effective surrogate model to mimic the input–output relationship as the Monte Carlo simulation–based on the mechanistic model might be computationally intensive. In this regard, the article presents a sparse regression method for structural reliability analysis based on the generalized polynomial chaos expansion. However, results from the global sensitivity analysis have justified that it is unnecessary to contain all polynomial terms in the surrogate model, instead of comprising a rather small number of principle components only. One direct benefit of the sparse approximation allows utilizing a small number of training samples to calibrate the surrogate model, bearing in mind that the required sample size is positively proportional to the number of unknowns in regression analysis. Therefore, by utilizing the standard polynomial chaos basis functions to constitute an explanatory dictionary, an adaptive sparse regression approach characterized by introducing the most significant explanatory variable in each iteration is presented. A statistical approach for detecting and excluding spuriously explanatory polynomials is also introduced to maintain the high sparsity of the meta-modeling result. Combined with a variety of low-discrepancy schemes in generating training samples, structural reliability and global sensitivity analysis of originally true but computationally demanding models are alternatively realized based on the sparse surrogate method in conjunction with the brutal Monte Carlo simulation method. Numerical examples are carried out to demonstrate the applicability of the sparse regression approach to structural reliability problems. Results have shown that the proposed method is an effective, non-intrusive approach to deal with uncertainty analysis of various structural systems.

This paper is devoted to the prediction and analysis of brake squeal under random uncertainty. This problem, which is a particular application of a wider issue, namely the stability of random parameter-dependent (RPD) nonlinear dynamical systems, is undertaken by using the non-intrusive generalized polynomial chaos (GPC) and Wiener-Haar expansions. The main objective is to assess the capacities of these meta-models within this framework. A reduced nonlinear non-equally damped, iso-damped and non-damped, disc/pad models are considered in this perspective in order to analyze the robustness of the proposed meta-models with respect to perfect and non-perfect mode coalescence. It turns out that the Wiener-Haar meta-model shows a more robust performance than GPC expansion and consequently offers a more reliable tool for the nonlinear stability analysis and thus for the prediction of brake squeal under parameter uncertainty.

In this paper, the design of a robust controller for systems with stochastic uncertainties is developed. The main idea is built on the generalized Polynomial Chaos theory, which transforms the stochastic dynamic model to an equivalent higher order deterministic dynamic model. The controller synthesis is formulated over the expanded system, taking in account the desired closed loop performances such as settling time and overshoot. The designed controller is then applied to the nominal and worst-case systems. Simulation results are presented to shed light on the efficiency and performance of the proposed controller.

The Moon–Rand systems, developed to model control flexible space structures, are systems of differential equations in R 3 of the form u = v , v = − u − u w , w = − λ w + f ( u , v ) . We give a partially positive answer to a recently conjecture for a special class of such systems, called the generalized polynomial Moon–Rand systems in the case when λ ∈ ( 0 , ∞ ) and f is either a homogeneous cubic, quartic, quintic or sextic polynomial.

In this paper, we present a low complexity bit-parallel Montgomery multiplier for $GF(2^{m})$ generated with irreducible Type C.1 pentanomials $x^{m}+x^{m-1}+x^{k}+x+1$ . Based on a combination of generalized polynomial basis (GPB) squarer and a newly proposed square-based divide and conquer approach, we can partition field multiplication into a composition of sub-polynomial multiplications and Montgomery/GPB squarings, which have simpler architecture and thus can be implemented efficiently. Consequently, the proposed multiplier roughly saves 1/4 logic gates compared with the fastest multipliers, while the time complexity matches previous multipliers using divide and conquer algorithms.

We obtain the exact values of the best (𝛼, β) -approximations for the classes K ∗ F of periodic functions K ∗ f such that f belongs to a given rearrangement-invariant set F and K is 2𝜋 -periodic kernel that does not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

In this paper, the element free Galerkin method is combined with the generalized polynomial chaos to quantify the uncertainties in the bending analysis of shear deformable plates with elastically restrained edges resting on a Pasternak elastic foundation with random system properties. The plate modules of elasticity, stiffnesses of the elastically restrained edges and the foundation stiffnesses are considered as random processes and are represented by using the Karhunen–Loeve expansion. It is shown that the results obtained by the presented method are in a very good agreement with the results of Monte Carlo simulations in spite of using low order polynomials in the generalized polynomial chaos expansion. Also the applicability and versatility of the presented method are demonstrated by solving numerical examples for various values of coefficient of variations, aspect ratio, thickness and several combinations of boundary conditions, different types of lateral loading and various values of stiffnesses of restrained edges and elastic foundation.